Experiment 2, Physics 2BL

Transcription

1 Experiment 2, Physics 2BL Deuction of Mass Distributions. Last Upate: Preparation Before this experiment, we recommen you review or familiarize yourself with the following: Chapters 4-6 in Taylor Moment of Inertia t-score 1. PHYSICS 1.1. Moments of inertia The three moments of inertia that are neee for this experiment are that of a hollow sphere with outer raius R an inner raius r: I hollow = 2 5 M (R5 r 5 ) (R 3 r 3 ) an that of a cyliner about its axis whether it is soli: or hollow: I soli = 1 2 MR2 I hollow = 1 2 M(R2 + r 2 ) The tricky part is that not all the elements will be irectly measurable. In the case of the racketball, you will not be able to break apart the ball in orer to measure the inner raius, r. You will use some clever methos to eucing these measurements from experiment. a ramp. The ball will start from rest at a height, h 1, at the first photogate, P G 1. We will measure the time, t, that it takes for the ball to get to travel a istance, x, when it arrives at a secon photogate, P G 2, which is at a height, h 2. As long as the ball starts at rest, the change in potential energy ue to the change in height, h = h 1 h 2, is converte into a final rotational an a final translational kinetic energy. mgh = 1 2 Iω2 f mv2 f (1) In orer for the experiment to work it is important for the ball to be rolling without slipping uring its journey own the ramp. This means that the the ramp must not be too steep or too slippery. Even though there is a constant force of friction on the ball, it oes no work on the ball because the frictional force oes not act over a istance. In other wors, the frictional force acts over zero isplacement. 1. The moment of inertia of a soli ball is I = 2 5 MR2. Write out the moment of inertia of a soli ball of raius r an ensity ρ in that has an outer coating of raius R (meaning a thickness of R r) that has a ensity ρ out, where ρ in an ρ out are constant Conservation of energy, rolling w/o slipping, rolling raius For the first part of this experiment we will be calculating the moment of inertia of a ball by rolling it own In our experiment there is a slight complication because the ball is not rolling on a flat surface but rather a square beam that is open on the top (See a cross of the ball sitting in this beam on page 1). This means that the rolling raius of the ball is smaller than its actual raius. We can calculate the rolling raius, R, by measuring the with of the beam an using the Pythagorean Theorem,

2 2 or by measuring the length of a full rotation of the ball an iviing by 2π. R = R 2 ( w 2 )2 2. If we rolle the ball own a ramp without a groove woul it roll faster or slower than the ball that is rolling own a square beam open on the top sie (A cross section of the ball sitting in such a beam can be seen in the iagram on page 1.)? In this case we are not concerne with resistance or whether the ball will follow a straight line but merely the effect of rolling raius on the spee of the ball. Explain your reasoning. Rolling without slipping gives us the conition ω = v R which can be plugge into the equation for conservation of energy in equation (1) an solve for I. Recall that R is the rolling raius an not the actual raius. I = M(R ) 2 [ 2gh v 2 f 1] Instea of measuring the velocity at the bottom of the ramp we will use the following formula that comes from motion uner constant acceleration. The final velocity, v f, ens up being twice the average velocity: v f = 2x t Which can be plugge into the expression for I: (2) I = M(R ) 2 [ ght2 1] (3) 2x2 3. What experimental conition no longer hols if you increase the ramp angle by a large amount or coat the ramp in oil? Briefly explain why. 2. METHODS FOR STATISTICAL ANALYSIS 2.1. Stanar Deviation an Stanar Deviation of the Mean Many times you will be making a series of measurements in orer to get a mean value. It is important to keep track of the ifference between the error on one of the measurements, which is the stanar eviation,, an the error (or stanar eviation) of the mean, σ x. The more measurements you make, the smaller the error on the mean becomes. Let s make sure we unerstan where this comes from. The equation for the mean of N values labele x i is a very simple function of N variables: x(x 1, x 2,..., x N ) = 1 N (x 1 + x x N ) Each one of the variables x i has a stanar eviation given by the following formula: = ( x xi ) 2 N 1 In orer to propagate the error from each of x i s to x we have to use the error propagation equation: σ x = [( x x 1 ) ] 2 + [( x x 2 ) ] [( x x N ) ] 2 Looking back on the equation for x we see that each of the x i s have the same partial erivative. x x i = 1 N Plugging everything in we get the following: σ x = ( N )2 + ( N ) ( σ x N )2 = N( N )2 Which reuces to a simple expression. σ x = (4) N 4. Suppose you are trying to measure the mass of an M&M an you measure 50 of them iniviually an obtain a mean value of 1.05g an a stanar eviation of.15g. What is the stanar eviation of the mean, σ m? How many M&M s shoul be masse in orer to get a σ m 0.01? Assume the stanar eviation oesn t change for more or less mass measurements. In other wors, use σ =.15g as the stanar eviation for the secon question. For extra practice see Taylor #4.18 an 4.26

3 Normal Distributions It is a fact of nature that many things that seem to have a uniform value, such as the with of the hairs on your hea or the size of the grains of san at the beach, when probe with precise measuring instruments actually ten to follow what we call a Gaussian, or Normal Distribution. That means that if we plot a histogram of these values most will be centere about a mean, with fewer an fewer at the extreme regions. The resolution of this curve improves if we ecrease the bin size an increase the sample size, but in general these curves have only two parameters. The mean, x, an the stanar eviation,. sie of the curve you will have to ivie these probabilities in half. t is known as the t-score an is efine as t = xi x, where x i is the value of the measurement in question, x is the mean of all the measurements, an is the stanar eviation of all the measure values of x. G(x) = 1 2πσ 2 x)2 exp[ (x 2σx 2 ] The useful thing about these curves is that when we normalize them (which means that we make the total area equal to 1) any portion of the area gives the probability of a value falling with that part of the curve. There is always a 68% probability of a value falling within 1σ of the mean, 95.4% probability of a value falling within 2σ of the mean, an 99.7% probability of a value falling within 3σ of the mean. For example if i tol you that the the average NBA basketball player was 75 ± 3 inches. If I foun a sample of 100 players, I woul expect 68 of them to be between 6 an 6 6. There shoul be about 16 players above 6 6 an 16 players uner 6. But what if we have a value that is a generic tσ away from the mean an we want to fin the likelihoo of our getting that value? All we have to o is fin the area uner the curve that is less than tσ away from the mean. To save you time, these integrals are alreay calculate for you in a chart, copie from p.287 in Taylor. The arrows in the table below show you how to rea the chart for when you want to fin the probability that a value is 1.47σ away from the mean. One tricky point to keep in min is whether the probability (area uner the curve) that you are intereste in refers to both the left an right of the curve or just one sie. If you are only intereste in one This chart, copie from p.287 in Taylor, gives the percentage probability that a given ata point will be within tσ of the mean. 5. Use the chart to fin the probability that a given ata point will be 1.52σ away from the mean? Suppose, for some set of ata, the mean is 5.3 an the stanar eviation is 0.4. What is the probability of getting a value larger than 6.1? Or a value smaller than 4.9? For extra practice see Taylor #4.14, 5.20, 5.36

4 Chauvenet s Criteria for Rejection of Data Suppose you have one ata point that is very far away from the mean an you want to know whether it is justifiable to throw it out? In most cases it is better to not throw away ata because you might be biase about the results of your ata. It is always better to simply take more ata to ecrease the effect of faulty ata points. Only consier throwing out a point if you feel like there was some error in the measurment collection. Again, a safer alternative is to just take more ata points until the suspicious point oes not contribute to the final results. Here is the Chauvenet rule-of-thumb for rejecting ata. t suspect = x suspect x n = N (1 P rob(t suspect )) 0.5 If we make N measurements an we want to challenge the valiity of x suspect we can only toss it out if n 0.5. Here, the P rob(t suspect ) is foun in table A. For extra practice see Taylor #6.4 WARNING: t is a variable commonly use to represent t-scores an time. Don t confuse them! When the variable t is use in these guielines, it shoul be clear, base on the context, whether we are speaking of the t-score or time Drawing a Histogram It can be very useful to convert a set of ata into a histogram. This allows you to see which points might be suspiciously off an wether the stanar eviation looks reasonable an is not being skewe by faulty ata points. Here is the proceure for rawing a histogram by han. 1. Determine the range of your ata by subtracting the smallest number from the largest one. 2. The number of bins shoul be approximately N an the with of a bin shoul be the range ivie by N. 3. Make a list of the bounaries of each bin an etermine which bin each of your ata points shoul fall into. 4. Draw the histogram. The y axis shoul be the number of values that fall into each bin. 5. Sometimes this proceure will not prouce a goo histogram. If you make too many bins the histogram will be flat an too few bins will not show the curve on either sie of the maximum. You might nee to play aroun with the number of bins to prouce a better histogram. 3. EXPERIMENTAL PROCEDURE, RACKETBALLS 3.1. Initial Measurements Set up the ramp an photogates accoring to the iagram on the first page. You shoul tape the photogates to the table. You can fin a more complete iagram of the set up in the lecture slies. Your TAs shoul give you a short emonstration of the setup before you begin. You will nee to measure the inner with of the ramp, w, the length of the ramp between the photogates, x, an the heights of the photogates, h 1 an h 2. Also, you will nee to choose a racquetball. There may be ifferent brans an colors of raquetballs in the lab. Which ever bran/color you choose, you must use that same kin of raquetball throughout the experiment! You also nee to measure the mass of the ball, M an its raius R. You may use whatever tools you like to make these measurements, just be sure to recor the associate uncertainties an units with each measurement. Finally, calculate the rolling raius, R One ball N times Set the photogates to PULSE moe, toggle switch to ON, an the black switch to 1ms. A crucial part of this experiment is releasing the ball consistently. The ball must begin with zero kinetic energy an from the same position for each trial. To o this, it is useful to tape a pen or pencil to the track. The pen/pencil shoul be perpenicular to the track an place before the first photogate sensor such that, when the ball rests against the pen/pencil, the ball sits just before the photogate sensor (The photogate shoul not be triggere when the ball is resting against the pen/pencil, but it will be triggere immeiately after you release the ball.). Acquiring goo ata in this experiment is ifficult an epens highly on your ability to release the ball consistently between trials. Before you begin your measurements, it is recommene that you practice releasing the ball. You shoul o at least 20 practice trials. If you on t conuct practice trials, it is likely you will obtain an imaginary number for σ t(man) (efine below), which is incorrect. After you complete your practice trials, begin measuring/recoring the time t for the ball to roll own the ramp. Do this N times. Normally you woul choose the number of trials N, however, in orer to acquire goo ata, you nee to o AS MANY TRIALS AS POSSI- BLE. N will epen on the number of balls available. You shoul be able to o at least 30 trials, but if there

5 5 are more than 30 balls, it is recommene that the number of trials you conuct equals the number of balls avaliable. Now, calculate t an the error in the measurement of t, σ t(meas) using equation (4). Use equation (3) to calculate I ± σ I. Remember, if you are using t in your etermination of I, then you must use the uncertainty associate with t, i.e. σ t, when calculating σ I. 6. Suppose you roll the ball own the ramp 5 times an measure the rolling times to be [3.092s, 3.101s, 3.098s, 3.095s, 4.056s]. Accoring to Chauvenet s criterion, woul you be justifie in rejecting the time measurement t = 4.056s? Show your work. 7.You are conucting the racquetball rolling experiment. During one of the trials your lab partner bumps into the table. Shoul you keep the ata from this trial? Explain your reasoning. 1 N σ. How oes it compare to the esire result of 1% or less? Finally, etermine what the value of N shoul be so that the uncertainty in σ T-SCORE ADDENDUM Earlier we expresse t-score as t = xi x. However, this form of t-score assumes x has zero uncertainty. A more general way of writing t-score woul be the following t = x 1 x 2 σ 2 x1 + σ 2 x 2 where x 1 an x 2 are two quantities you are comparing an both have an uncertainty, 1 an 2 respectively. Note that this expression can be reuce to our previous expression if we assume one of the values has zero uncertainty. The important point is that the general t-score equation shoul be use when comparing two values that each have non-zero uncertainties N balls one time Now collect N balls an recor the time for each to roll own the ramp. From this experiment you can etermine the uncertainty in t that is ue to the combine effects of measurement error an manufacturing error, which we call the total error σ t (total). This is simply the stanar eviation of the time ata collecte in this experiment. Calculate t an σ t (total) Calculation of σ t (man), Percent error for, an etermination of N Use the following formula to calculate the error ue to manufacturing: σ t (man) = σ t (total) 2 σ t (meas) 2 Now the percent variation in the thickness of the racketball,, is given by the following relationship: σ = 27σ t(man) t This relationship is obtaine by using typical values for equations (3) an the moment of inertia of a hollow sphere an then making small eviations from those typical values to fin how sensitive errors in are to errors in I an how sensitive errors in I are to errors in t. See the lecture slies for a etaile erivation of this equation. The goal of this experiment is to etermine σ to an accuracy of 1%. Calculate the uncertainty in σ, equal to